Problem 5 Solve the given equations for \(... [FREE SOLUTION]

Chapter 3: Problem 5

Solve the given equations for $0^{\circ} \leq x<360^{\circ} .$ $$ \csc ^{2} x+4 \csc x-7=0 $$

Short Answer

Expert verified

The solutions to the equation in the given domain $0^{\circ} \leq x < 360^{\circ} $ are $x = \sin^{-1}(1/7), 180^{\circ}-\sin^{-1}(1/7), $ and $ 270^{\circ} $.

Step by step solution

Rewrite the Equation

Write the given equation in terms of sine. Since $ \csc x = \frac{1}{\sin x} $, the equation becomes $\frac{1}{\sin^2 x} + \frac{4}{\sin x} - 7 = 0$. Multiply all terms by $ \sin^2 x $ to eliminate the denominator, which yields: $1 + 4 \sin x \sin^2 x - 7 \sin^2 x = 0$

Simplify and Factorize

Rewrite the equation as a quadratic equation in terms of $ \sin x $, as -7 sin^2 x + 4 sin x + 1 = 0, and then factorize it. To factorize, we find two numbers that add up to 4 and multiply to -7. These numbers are -1 and -7, so factoring gives: -(7 sin x - 1)(sin x + 1) = 0.

Solve for $ x $

Set each factor equal to zero, yielding the following equations: $7 \sin x - 1 = 0$ and $ \sin x + 1 = 0 $. Solve for $ x $ in both equations.

Find $ x $ in the Given Domain

By solving the equations, $ x = \sin^{-1}(1/7) $ or $ 180^{\circ}-\sin^{-1}(1/7) $ from the first equation, and $ x = \sin^{-1}(-1) $ from the second equation. By checking the available range, the solutions within the range $ 0^{\circ} \le x < 360^{\circ} $ are $ x = \sin^{-1}(1/7), 180^{\circ}-\sin^{-1}(1/7), $ and $ 270^{\circ} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function

The cosecant function, often represented as $ \csc x $, is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function, which means $ \csc x = \frac{1}{\sin x} $. Understanding this relationship is crucial in solving trigonometric equations involving cosecant.

The cosecant function is undefined for values of $ x $ where $ \sin x = 0 $ because division by zero is undefined.
The function is periodic, repeating every $ 360^{\circ} $, just like the sine function.
In practical terms, $ \csc x $ represents the ratio of the length of the hypotenuse to the length of the opposite side in a right-angled triangle.

When dealing with equations like $ \csc^2 x + 4 \csc x - 7 = 0 $, it's often useful to express everything in terms of sine to simplify the problem-solving process, as shown in the provided steps for solving the original exercise.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, generally given by $ ax^2 + bx + c = 0 $, where $ a, b, $ and $ c $ are constants, with $ a eq 0 $. In the context of trigonometric equations, you might see a quadratic form like in $ -7 \sin^2 x + 4 \sin x + 1 = 0 $.

Quadratics can be solved by factoring, completing the square, or using the quadratic formula $ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} $.
Factoring involves rewriting the polynomial as a product of two binomials.
In trigonometric problems, we set each factor equal to zero to find the possible $ x $ values.

It鈥檚 a systematic process that relies on identifying the right factors that can simplify the expression, just as with the equation given in the initial problem.

Sine Function

The sine function, $ \sin x $, is a basic yet pivotal trigonometric function that relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. Its range of values is between $-1$ and $1$. This characteristic plays a crucial role in solving trigonometric equations.

The sine function is periodic, repeating every $ 360^{\circ} $.
It's positive in the first and second quadrants and negative in the third and fourth.
This behavior helps to determine where potential solutions fall within a given range.

In the context of solving the equation $ \csc^2 x + 4 \csc x - 7 = 0 $, converting the terms into sine allows for easier manipulation and solution finding.

Angle Solutions

Finding angle solutions in trigonometric equations involves identifying the values of $ x $ that satisfy the equation within a specific range or domain. For equations like $ \csc^2 x + 4 \csc x - 7 = 0 $, solution steps typically include the following:

Converting the equation into a form that can be solved using standard algebraic techniques, like factoring or using the quadratic formula.
Identifying all possible solutions within the given range, often $ 0^{\circ} \leq x < 360^{\circ} $.
Considering the general solution from trigonometric identities and properties.

For our specific problem, the solutions included angles where $ \sin x = \frac{1}{7} $ and where $ \sin x = -1 $. Checking these within the range ensured they fell within $ \left[0^{\circ}, 360^{\circ}\right) $, leading to the final answers of $ \sin^{-1}(1/7), 180^{\circ}-\sin^{-1}(1/7), $ and $ 270^{\circ} $.

91影视

Solve the given equations for \(0^{\circ} \leq x<360^{\circ} .\) $$ \csc ^{2} x+4 \csc x-7=0 $$

Short Answer

Step by step solution

Rewrite the Equation

Simplify and Factorize

Solve for \( x \)

Find \( x \) in the Given Domain

Key Concepts

Cosecant Function

Quadratic Equation

Sine Function

Angle Solutions

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